By Alessio F., Montecchiari P.
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Additional info for A note on the multiplicity of entire solutions in R2 for a class of almost periodic Allen-Cahn type equations
1 The discovery of irrational numbers constitutes the first foundational crisis of mathematics. 2 Apart from the contribution of Zeno's B Fragment 3 to our understanding of the nature of the spatial whole-parts relationship, we find statements in Anaxagoras regarding the nature of spatial continuity which are still actual today. He says: In that which is small there is no smallest, since there always exists something smaller. 3). 6). 9). e. inwardly) as infinitely divisible spatial extension. One of the categories which Aristotle distinguishes is quantity.
A) Aristotle's objections against the actual infinite Aristotle's first objection, namely that the whole could not in the case of the actually infinite exceed the part in magnitude, had already been used by Bolzano (analogously to Galileo) exactly as criterion for infinite sets: a set is infinite if and only if the whole set can be mapped one-to-one with a true subset thereof. In this characterization lies the answer to the first part of Aristotle's second objection, the objection namely that it is impossible to count the infinite.
There is a further remarkable side to this result. In his original proof of 1874 (1962:115-118) Cantor first proved that all real algebraic numbers are 1 Cf. Heyting (1971:40), Fraenkel et al (1973:256,272), and Fraenkel (1928:239 note 1). 37 denumerable. 2 By 1895 Cantor discovered that his set theory contained anomalies. g. the proposition that for every set A of ordinal numbers an ordinal number exists which is greater than every ordinal number contained in the set. Consider however the set W of all ordinal numbers.
A note on the multiplicity of entire solutions in R2 for a class of almost periodic Allen-Cahn type equations by Alessio F., Montecchiari P.